Lesson 12-1: Size Changes on a Coordinate Plane (Day 121)

Chapter 12 of the U of Chicago text is called "Similarity." I've written a little on similarity lately, since students in my new district have just finished Chapter 7 of the Glencoe text, also on similarity.

This is also the chapter from which I started posting last year, after I left my old job. Yes, it's now officially a year since I left. Because of this, I can start reblogging my lessons from last year, rather than from two or three years ago.

We've now officially covered every lesson in the U of Chicago text, since last year we covered all of Chapters 12 through 15, and this year we cover the first eleven chapters. I no longer include Lessons 8-1 through 8-3 in the curriculum, but I did two years ago (the first three days of March 2016).

Of course, some might argue that this is a cop-out. Yes, last week I had a Lesson 11-6 worksheet, but it was a two-dimensional graphing activity when 11-6 in the text is three-dimensional graphing. And often other lessons I post are derived from other sources besides the text. Indeed, Lesson 12-1 from last year -- that I reblog this year -- is from an activity I found online.

Lesson 12-1 of the U of Chicago text is called "Size Changes on a Coordinate Plane." In the modern Third Edition of the text, size changes on a coordinate plane appear in Lesson 3-7. Yes, Chapter 12 is officially the same in both editions, but for some reason, the new edition introduces transformations involving size ("dilations") very early in the text. Beginning with the old Lesson 12-3, most of the old Chapter 12 material does indeed appear in the new Chapter 12 as well.

This is what I wrote last year -- yes, last year -- about today's lesson:

In the past, I skipped over Lesson 12-1. This is because I was mainly concerned with circularity -- dilations are used to prove some of the properties of coordinates, but right in this lesson, coordinates are used to prove the properties of dilations.

But last year, I was fed up with juggling the order of the U of Chicago text (and this year, I got in trouble trying to juggle the Illinois State text as well). This year I want to stick to the order as intended by the authors of the U of Chicago text. And furthermore, we've seen that the actual dilation problems on the PARCC and SBAC involve performing dilations on a coordinate plane -- not using dilations to prove properties of coordinates! So Lesson 12-1 is more in line with PARCC and SBAC.

Here is the main theorem of Lesson 12-1 along with its coordinate proof:

Theorem:

Let S_k be the transformation mapping (x, y) onto (kx, ky).

Let P' = S_k(P) and Q' = S_k(Q). Then

(1) Line P'Q' | | line PQ, and

(2) P'Q' = k * PQ.

Proof:

Let P = (a, b) and Q = (c, d) be the preimages.

Then P' = (ka, kb) and Q' = (kc, kd).

(1) Line P'Q' is parallel to line PQ if the slopes are the same.

slope of line P'Q' = (kd - kb) / (kc - ka) = k(d - b) / k(c - a) = (d - b) / (c - a)

slope of line PQ = (d - b) / (c - a)

Thus line PQ | | line P'Q'.

(2) The goal is to show that P'Q' = k * PQ.

From the Distance Formula,

PQ = sqrt((c - a)^2 + (d - b)^2).

Also from the Distance Formula,

P'Q' = sqrt((kc - ka)^2 + (kd - kb)^2)

        = sqrt((k(c - a))^2 + (k(d - b))^2)      (Distributive Property)

        = sqrt(k^2(c - a)^2 + k^2(d - b)^2)    (Power of a Product)

        = sqrt(k^2((c - a)^2 + (d - b)^2))       (Distributive Property)

        = sqrt(k^2)sqrt((c - a)^2 + (d - b)^2) (Square Root of a Product)

        = ksqrt((c - a)^2 + (d - b)^2)              (Since k > 0, sqrt(k^2) = k)

        = k * PQ                                             (Substitution) QED

At the end of this post, it's back to posting worksheets based on the U of Chicago text. This time, I post an activity from last year where students dilate cartoon characters. This activity makes more sense this year than last year since it requires using coordinates.

This lesson could actually help my eighth graders as well. We were supposed to cover dilations earlier but we ran out of time. For that matter, slope and the Distance Formula are also part of the eighth grade curriculum. I wouldn't make eighth graders perform the two preceding proofs with so many variables, but specific numerical examples are within the reach of eighth graders.

Returning to 2018, okay, I feel guilty about not posting any true Lesson 11-6 exercises. So here's what I'll do -- I include a new worksheet with Question 11 from Lesson 12-1. This question is on dilations in three dimensions. These work exactly how we expect them to work. Students also practice graphing in space as well as use a Distance Formula in Three Dimensions. So this question incorporates Lesson 11-6 as well.

Now I can truly say that I've covered every lesson in the text here on the blog.